Quantum size effects in ultra-thin YBa2Cu3O7 − x films

The d-wave symmetry of the order parameter with zero energy gap in nodal directions stands in the way of using high-temperature superconductors for quantum applications. We investigate the symmetry of the order parameter in ultra-thin YBa2Cu3O7 − x (YBCO) films by measuring the electrical transport properties of nanowires aligned at different angles relative to the main crystallographic axes. The anisotropy of the nanowire critical current in the nodal and antinodal directions reduces with the decrease in the film thickness. The Andreev reflection spectroscopy of the nanoconstrictions shows the presence of a thickness-dependent energy gap that does not exist in bulk YBCO. We find that the thickness-dependent energy gap appears due to the quantum size effects in ultra-thin YBCO films that open the energy gap along the entire Fermi surface. The fully gapped state of the ultra-thin YBCO films makes them a promising platform for quantum applications, including quantum computing and quantum communications.

Theoretically, it was proposed that the scattering 11 , disorder 12 , or large superconducting phase fluctuations 13,14 may result in the fully gapped state of the d-wave superconductor. All this evidence gives hope that it is feasible to build fully gapped devices from the high-T c superconductors which can be used for quantum applications. However, there is still no recipe for preparing the fully gapped state in the optimally-or overdoped d-wave superconductor.
Following the observation of the fully gaped state in the ultra-thin YBCO nanowires 7   characteristics. While the Andreev spectra of nanoconstriction provide information on the absolute magnitude of the energy gap in a wide range of angles.

Study of order parameter symmetry with nanowires
We fabricate the 530-nm-long nanowires with an effective width of 70-100 nm from the optimally-doped ultra-thin YBCO films with a thickness d in the range of 7 -10.5 nm (6-9 unit cells (u.c.)). The c-axis-oriented YBCO films have a roughness of ± 1 u.c. 15 . On each substrate, we pattern thirteen nanowires in the nodal and antinodal directions as it is schematically shown in Figure 1b. The representative SEM micrograph of the nanowire shaped by two cuts using focused ion beam (FIB) is demonstrated in Figure 1c. The nanowires are covered with a 15-nm-thick gold layer deposited in situ to get the same boundary conditions as in nanoconstrictions, presented below, where the gold layer prevents overheating at high voltage biases.
We measure the current-voltage (IV) characteristic of the current-biased nanowire at a temperature T = 4.2 K to determine the critical current I c . The IV curve demonstrates a voltage switching at a current above the critical current with the small current hysteresis (see the Supplementary information). The critical currents of the nanowires oriented along one antinodal direction are higher than those of the nanowires oriented along the perpendicular antinodal direction. Since the energy gap Δ in YBCO is larger in the baxis direction than in the a-axis direction 16 , we assign the nanowires with the lower and higher critical currents to the a-and b-axis directions, respectively.
The critical current density is calculated as J c =I c /W eff d eff using the effective width W eff = W -140 nm instead of geometric width W, as outlined in our previous work 17 , and the effective thickness d eff = d -2 u.c. taking into account the non-superconducting YBCO layer at the YBCO/SrTiO 3 interface. The thickness dependences of the average critical current density <J c0 > in the a-axis direction (0°), <J c90 > in the b-axis direction (90°), and <J c45 > in the nodal direction (45°) are plotted in the inset in Figure 2. The normalized <J c > values along nodal and antinodal directions are shown in Figure 2. The average critical current density <J c > in the antinodal directions significantly reduces with the decrease of the film thickness preserving the same <J c0 >/<J c90 > ratio of 0.7 for the 7-9-u.c.-thick nanowires. This ratio is the same as the in-plane anisotropy Δ a /Δ b in the single-crystal YBCO samples 16 . For the thinnest 6-u.c.-thick nanowires, as shown in Figure 2, we observe a stronger reduction of the critical current density in the a-axis direction compared to that in the b-axis direction which is evidence for the suppression of the superconducting energy gap in the a-axis direction.
The nodal critical current density of the thickest nanowire is significantly smaller than those in the antinodal directions, as expected in the case of the d x2-y2 -wave pairing symmetry. However, the <J c45 >/<J c0 > and <J c45 >/<J c90 > ratios are not as small as in bulk samples, where they are about a few percent 18,19 .
Remarkably, the <J c45 > thickness dependence is different from those in the antinodal directions.
The critical current density in the nodal direction doesn't demonstrate a monotonic thickness A superconducting constriction can be modeled as a superconductor-normal metal-superconductor (SNS) junction where the voltage at currents above the critical current is developed across the dissipative neck region 21,22 . Within the framework of the simplified theoretical approach to SNS junctions, each quasiparticle undergoes multiple Andreev reflections (MAR) before it is scattered or leaves the pair potential well. If a quasiparticle undergoes n Andreev reflections (AR), then ne charges are transferred through the NS boundary in addition to the initial one, and the SNS junction current has is enhanced due to AR. Here n is an integer and e is an electron charge. As consequences of MAR, the IV curves and especially the conductance curves show many nonlinear structures [23][24][25] .  To overcome this problem, we measure additionally 100-nm-wide YBCO nanoconstrictions covered in situ by a 15-20-nm-thick gold layer. The 100-nm-wide constrictions are wide enough to avoid electrical transport across the degraded YBCO in the constriction area. The gold layer removes the heat from the constriction area and prevents the normal-state region in the constriction from expanding. The Au/YBCO nanoconstrictions are fabricated with electron-beam lithography and ion-beam etching, as described in the Methods. A representative SEM micrograph of the nanoconstriction is shown in Figure 1d.
Representative differential conductance curves of 100-nm-wide and 5-9-u.c.-thick Au/YBCO nanoconstrictions are shown in Figure 4a. Except for the thinnest nanoconstrictions, the differential conductance demonstrates steps similar to those observed for the ultra-narrow constriction without gold capping layer. We clearly identify the conductance steps at high voltages which correspond to the energy gaps in the bulk YBCO. These steps, which appear at voltages V = Δ a /e ≈ 30 mV and V = Δ b /e ≈ 45 mV, are We plot the positions of the steps found at the differential conductance of the 100-nm-wide Au/YBCO nanoconstrictions of various thicknesses in Figure 4b. The region corresponding to the Δ a /e and Δ b /e are colored in blue and green, respectively. The conductance steps at voltages around 20 mV can be assigned as 2Δ a /3e or Δ b /2e. An accurate MAR analysis is intrigue because Δ b ≈ 1.5Δ a , therefore, a series of MAR harmonics originating from Δ a intersects with a series of MAR harmonics originating from Δ b . However, the low-voltage conductance steps shown in Figure 4a by black arrows belong neither to 2Δ a /ne series nor to 2Δ b /ne series. These conductance steps are caused by the energy gaps which are not inherent to the bulk YBCO. The energies of these low-energy gaps are smaller for the thicker films and higher for the

Quantum size effects in nanoscale YBCO films
The (1/d) 2 -type dependence of the energy gap on the device size is typical for the quantum size effects (QSE) which occur when at least one dimension of the device is comparable with the Fermi wavelength λ F = h/mv F , where h is the Planck constant, m is the charge carrier mass, and v F is the Fermi velocity. QSE are well known for semiconducting nanodevices because of the large Fermi wavelength 28  The thickness dependence of the new gap in the ultra-thin YBCO films is in good quantitative agreement with the well-known expression for the confinement energy in the quantum well E g = [ħπ/d] 2 /2m (1) which is shown in Figure 4b by dashed lines. We plot two E g -dependencies which account for the film roughness of 1 u.c. and nonsuperconducting layer at the YBCO-substrate interface. The upper and the lower dashed lines correspond to E g = [ħπ/(d-3 u.c.)] 2 /2m e and E g = [ħπ/(d-2 u.c.)] 2 /2m e , respectively. The changes in the superconducting properties become even more drastic when the magnitude of the confinement gap approaches the value of the superconducting energy gap, as shown at the rightmost steps in Figure 1a.
The superconducting order parameter is strongly dependent on the number of the single-electron states inside the Debye "window" around the Fermi level 45 . The increasing confinement energy gap first reduces the number of states that can be used for the quasiparticle pairing resulting in the decrease of the critical current density, as it is confirmed by the experimental results for the nanowires in Figure 2. When the magnitude of the confinement gap approaches the value of the Debye energy, which is ħω D ≈ 40 meV ≈ Δ b in YBCO 46 , the critical current density is strongly suppressed and the conductance behavior is dominated by the confinement energy gap as it is observed for the YBCO nanoconstrictions with the total and effective thickness of 5 and 3 u.c., respectively.
The fully gapped state due to QSE explains many of the unusual experimental results with nanoscale cuprate superconductors which contradict the d-wave symmetry. Finally, we compare our findings with the preceding experimental results. In our previous work, we estimated the superconducting energy gap

Conclusions
We investigate the superconducting properties of ultra-thin YBCO films and find that quantum size effects open the energy gap in the nodal direction and turn the d-wave superconductor into the fully gapped state with the magnitude of the nodal energy gap given by the confinement energy E g = (ħπ/d) 2 /2m e . The nanoscale YBCO film can be considered as a quantum-engineered superconductor where the superconducting gap is controlled by quantum effects. The fully gapped state paves the way for nanoscale d-wave superconductors towards many quantum applications including quantum computing and singlephoton detection.

Methods
Nanostructure fabrication. YBCO nanowires and nanoconstrictions were fabricated from a 5.9-10.5 nm (5 -9 unit cell) thick YBCO film deposited on a TiO 2 -terminated (100) single-crystal SrTiO 3 substrate by dc sputtering at high (3.4 mbar) oxygen pressure. One unit cell corresponds to the YBCO lattice parameter in c-direction which is 1.168 nm. YBCO deposition followed a procedure that is described elsewhere 15 . After that YBCO films were in situ capped either with a 6-nm-thick amorphous YBCO layer deposited at room temperature in the case of the ultra-narrow nanoconstrictions or by the 15-20 nm-thick gold layer deposited at T = 100°C by magnetron dc sputtering in argon. We do not see any effect of the gold layer on the nanowire or nanoconstriction superconducting properties. The 100-nm-thick Au contact pads were deposited ex situ using room temperature dc magnetron sputtering with a shadow mask. Following contact pad deposition, the nanostructures were fabricated in a two-stage process. In the first stage, 5μm-wide microbridges oriented along [100] and [010] crystallographic axes of the SrTiO 3 substrate were patterned using an optical UV contact lithography with a PMMA resist and Ar ion beam etching or wet chemical etching in a Br-Ethanol solution. In the second stage, the 530-nm-long nanowires and the 40-50nm-long nanoconstrictions were fabricated across the microbridges with two cuts made with FIB milling using an Au/PMMA protective layer or with the ion-beam etching in argon through the 80-nm-thick CSAR62 resist mask patterned by the electron-beam lithography. More details on the patterning process can be found in ref. 17.
Experimental setup. The experimental setup was based on a liquid helium storage Dewar insert filled with He exchange gas. The temperature above 4.2 K was maintained with the resistive heater controlled by a Lake Shore temperature controller 335. We used battery-operated low-noise analog electronics to sweep the bias current and amplify the voltage across the nanostructure. The differential resistance of the nanostructure was measured with lock-in amplifier 7265 (Signal Recovery) at 10 kHz modulation frequency. Figure